© Ting-Ting Sun et al. 2010
Received: 6 October 2009
Accepted: 19 January 2010
Published: 28 January 2010
The nonlocal boundary value problem, with -Laplacian of the form , has been considered. Two existence criteria of at least one and three positive solutions are presented. The first one is based on the Four functionals fixed point theorem in the work of R. Avery et al. (2008), and the second one is based on the Five functionals fixed point theorem. Meanwhile an example is worked out to illustrate the main result.
Due to the unification of the theory of differential and difference equations, there have been many investigations working on the existence of positive solutions to boundary value problems for dynamic equations on time scales. Also there is much attention paid to the study of multipoint boundary value problem with -Laplacian; see [1–10].
In , the author discussed the positive solutions of a -point boundary value problem for a second-order dynamic equation on a time scale
Zhao and Ge  considered the following multi-point boundary value problem with one-dimensional -Laplacian:
By using the Four functionals fixed point theorem and Five functionals fixed point theorem, we obtain the existence criteria of at least one positive solution and three positive solutions for the BVP (1.3). As an application, an example is worked out finally. The remainder of this paper is organized as follows. Section 2 is devoted to some preliminary discussions. We give and prove our main results in Section 3.
The basic definitions and notations on time scales can be found in [12, 13]. In the following, we will provide some background materials on the theory of cones in Banach spaces. For more details, please refer to [14, 15].
A map is said to be a nonnegative continuous concave functional on a cone of a real Banach space if is continuous and for all and . Similarly, we say that the map is a nonnegative continuous convex functional on a cone of a real Banach space if is continuous and for all and .
The following lemma can be found in .
Lemma 2.3 (four functionals fixed point theorem).
We are now in a position to present the Five functionals fixed point theorem (see ). Let be nonnegative continuous convex functionals on and nonnegative continuous concave functionals on . For nonnegative numbers and define the following convex sets:
Lemma 2.4 (five functionals fixed point theorem).
Define a cone
3. Main Results and an Example
In order to complete the proof of Theorem 3.1, we first need to prove the following lemma.
Applying the Arzelà-Ascoli theorem on time scales , one can show that is relatively compact.
From the definition of , we know that on . This shows that each subsequence of uniformly converges to . Therefore the sequence uniformly converges to . This means that is continuous at . So, is continuous on since is arbitrary. Thus, is completely continuous. This completes the proof.
Proof of Theorem 3.1.
Thus (i) and (ii) in Lemma 2.4 hold.
This completes the proof.
This work is supported by the Natural Science Foundation of Ludong University (24200301, 24070301, 24070302), Program for Innovative Research Team in Ludong University, and a Project of Shandong Province Higher Educational Science and Technology Program.
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